Using the GDC to calculate n = 10, 20, 50 the result was:
n = 10
M10 = = =
n = 20
M20 = = =
n = 50
M50 = ≈ =
By calculating the values that were given in question number 1, I noticed a pattern that occurs from one value of n to the other. The 2 inside the matrix are always raised to the value of n in Mn. The result gathered was that in entries ‘a’ and ‘d’ which are number greater than one, follows this pattern:
Mn =
To show whether my assumption is right or wrong let us try a couple more value calculating it manually and then comparing it using the graphics calculator.
Let us use values for n such as 25, 30, 40, and 45.
So for all values of n calculated using the general expression in the first question matches to the result when n is calculated using GDC. Therefore the general expression of Mn = is valid.
QUESTION 2
Consider the matrices P = and S =.
P2 = = = 2; S2 = = = 2
Calculate Pn and Sn for other values of n and describe any pattern(s) you observe.
Pn and Sn was calculated using the GDC for values such as n = 3, 4, 5, 6, 7, 8, 9, 10
I started using the GDC by inserting matrices P and S to the memory slot. For P the matrix slot that was used was [B] and for S I used matrix slot [C]. I specifically entered those matrices in a 2 x 2 dimension to suit the purpose. After storing the matrix in the memory I accessed it using the MATRIX, NAMES submenu and raise it to the value of n that I was calculating.
The results I got by calculating P in the GDC was
For the second part I will calculate Sn using n as 3, 4, 5, 6, 7, 8, 9, and 10.
By calculating the different values of n for Pn and Sn a specific pattern could be observed, which is:
Xn = 2(n1)
X in this general expression corresponds to the matrix P or S, and R inside the matrix is 2 when P is used, and it is 3 when S is used. I will now try to recheck whether the general formula by using the general formula that I observed and comparing it to the results calculated using the GDC.
P5 = 2(51)= 24= 16
S5 = 2(51)= 24= 16
The general formula gave the same answer as to the result gathered using the GDC. That means the general formula is acceptable.
QUESTION 3
Now consider the matrices of the form.
Steps one and two contain examples of these matrices for k – 1, 2 and 3.
Consider other values of k, and describe any pattern(s) you observe.
Generalize these results in terms of k and n.
The matrices that were given in question 1, 2, and 3 were derived from the form. Example, if K was given to be 2, then the result would be, which was the question in number one. It is the same in number 2 where K for the matrix P is 2 and in S is 3 the result in correspondingly are and . So it could be said that the matrices that was given in the previous questions follow the matrix form of. So when K is 4, 5, 6, and 7 we get:
I am able to analyse that the difference within the entries are always 2 (i.e. 31 = 2, 86 = 2). Thus if the difference is the same, the will be the same as the previous:
Xn = 2(n1)
To make sure that my statement is true, I will try and use an example:
Take K as 5 for example. By using the matrix I would get the value of K= 5 as. Lets call this matrix of K=5 as Z.
By calculating manually: Z6=.
By using the GDC: Z6=
Therefore, we can say that the general formula of
Xn = 2(n1) is applicable with the matrix.
Again it is important to make clear what the symbols represent; the X in this equation would represent the symbol of the matrix related (such as P, S or M). The n would be the power in which the matrix is being raised to, and K is going to be the R of Matrix X. To make this concept perfectly clear I shall undertake another example:
Take these values into consideration:
K= 3 ; n = 2
K=3 = = =
Let us name this matrix Y. If Y was raised to n = 2, than we would use the equation of Xn = 2(n1).
Now remember that K=R thus R would be 3
Y2 = 2(21)= 2
This result matches with our results that we had counted in question 2 for S2.
Thus from all this calculation we have gathered the following:
 That K=R

That the general formula Xn = 2(n1). Can be used in the matrix .
Thus we can change R in our general formula with K to become
Xn = 2(n1)
QUESTION 4
Use technology to investigate what happens with further values of k and n. State the scope or limitations of k and n.
To find the limitation of this general formula I used the GDC to calculate some values regarding to K and n. The values include whole numbers, integers, fractions and irrational numbers. In this part I tried to compare between my manual calculation using the general formula and comparing it with my calculation using the GDC. In this testing I will keep my K=1 through out the testing to find the limitation for n. The results are as followed:
From the results above, it is evident that some types of numbers are not suitable when being calculated by the GDC. Those numbers include, work for negative whole numbers, fractions, negative fractions, decimals, negative decimals, surds, irrational numbers and numbers greater than 255 as proved using the GDC above.
However, the general formula does work when the value of n is a positive integer, a shown above using the value n = 0, 5 and 255. Thus it can be concluded that the general equation only works when the value of n is a positive integer.
On the other hand, any value of K is acceptable and can be used in this general formula as long as the entries inside the matrix has a difference of 2, which it follows the matrix formula of k+1 and k1. If the matrix has a difference of 2 (regardless of it being negative, positive, fractions, decimals, irrational, surds etc.) the general formula would work.